Suppose P is drawn from the Uniform[0, 1] distribution, and then conditional on P, another random variable X is drawn from a Bernoulli(P) distribution. (a) Use the tower law to compute P(P ≤ t, X = 1) where t ∈ [0, 1] is some constant value

The moment generating function of the discrete random variable X is given below: Which of the following is the probability P (X <4) of the random variable X? X kesikli raslantı değişkenin moment çıkaran fonksiyonu aşağıda verilmiştir: 1 Mx(t)=e 4 1 41 + e" + 12 22 - X raslantı değişkenin P(X<4) olasılığı aşağıdakilerden hangisidir? O A) 1/12 O B) 7/12 O C) 5/12 O D) 314 O E) 1/4

let X(n,p) denotes binomial random variable with parameters n and p. Show that X(n+1,p) >st ( greater than equal ) X(n,p)

[Bayes' Theorem, Discrete Random Variables] It is known that 90% of new cars produced by a certain company have on average À malfunctions per 1 km of covered distance. Besides this, it is known that remaining 10% of new cars were produced with a defect which increases the average malfunction rate to 2λ. The two types of cars are indistinguishable otherwise. When (a) Suppose that only a single malfunction has occured with a randomly selected car over 1,000 km of distance. What is the probability that the car is from the defective group, and how does it depend on λ? (b) For a randomly selected car, find a minimum distance covered with no malfunctions for the probability that the car is from the non-defective group is 95% or above (express the result as a function of X).